Morita Equivalence for Crossed Products by Hilbert C-Bimodules

We introduce the notion of the crossed product$A\rtimes _{X}Z$of a C*-algebra A by a Hilbert C*-bimodule X. It is shown that given a C*-algebra B which carries a semi-saturated action of the circle group (in the sense that B is generated by the spectral subspaces B0and B1), then B is isomorphic to the crossed product$B_{0}\rtimes _{B_{1}}Z$. We then present our main result, in which we show that the crossed products$A\rtimes _{X}Z$and$B\rtimes _{X}Z$are strongly Morita equivalent to each other, provided that A and B are strongly Morita equivalent under an imprimitivity bimodule M satisfying X⊗AM≃ M⊗BY as A - B Hilbert C*-bimodules. We also present a six-term exact sequence for K-groups of crossed products by Hilbert C*-bimodules.